Secondary Two Mathematics: An Integrated …...Circles and Other Conics 8.1 Classroom Task: 8.2 Getting Centered – A Solidify Understanding Task Complete the square to find the center

© 2013 Mathematics Vision Project | MVP

In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license.

Secondary Two Mathematics:

An Integrated Approach

Module 8

Circles and Other Conics

By

The Mathematics Vision Project:

Scott Hendrickson, Joleigh Honey,

Barbara Kuehl, Travis Lemon, Janet Sutorius www.mathematicsvisionproject.org

In partnership with the

Utah State Office of Education

1

http://www.mathematicsvisionproject.org/


In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license.

Module 8 – Circles and Other Conics

Classroom Task: 8.1 Circling Triangles (or Triangulating Circles) – A Develop Understanding Task Deriving the equation of a circle using the Pythagorean Theorem (G.GPE.1) Ready, Set, Go Homework: Circles and Other Conics 8.1 Classroom Task: 8.2 Getting Centered – A Solidify Understanding Task Complete the square to find the center and radius of a circle given by an equation (G.GPE.1) Ready, Set, Go Homework: Circles and Other Conics 8.2 Classroom Task: 8.3 Circle Challenges – A Practice Understanding Task Writing the equation of a circle given various information (G.GPE.1) Ready, Set, Go Homework: Circles and Other Conics 8.3 Classroom Task: 8.4 Directing Our Focus– A Develop Understanding Task Derive the equation of a parabola given a focus and directrix (G.GPE.2) Ready, Set, Go Homework: Circles and Other Conics 8.4 Classroom Task: 8.5 Functioning with Parabolas – A Solidify Understanding Task Connecting the equations of parabolas to prior work with quadratic functions (G.GPE.2) Ready, Set, Go Homework: Circles and Other Conics 8.5 Classroom Task: 8.6 Turn It Around – A Solidify Understanding Task Writing the equation of a parabola with a vertical directrix, and constructing an argument that all parabolas are similar (G.GPE.2) Ready, Set, Go Homework: Circles and Other Conics 8.6

2


In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license

8.1 Circling Triangles (or Triangulating Circles) A Develop Understanding Task

Using the corner of a piece of colored paper and a ruler, cut a right triangle

with a 6” hypotenuse, like so:

Use this triangle as a pattern to cut three more just like it, so that you have a total of four congruent

triangles.

1. Choose one of the legs of the first triangle and label it x and label the other leg y. What is

the relationship between the three sides of the triangle?

2. When you are told to do so, take your triangles up to the board and place each of them on

the coordinate axis like this:

Mark the point at the end of each hypotenuse with a pin.

© 2

01

3 h

ttp

://f

lic.k

r/p

/5m

EpJm

3



3. What shape is formed by the pins after the class has posted all of their triangles? Why

would this construction create this shape?

4. What are the coordinates of the pin that you placed in:

a. the first quadrant?

b. the second quadrant?

c. the third quadrant?

d. the fourth quadrant?

5. Now that the triangles have been placed on the coordinate plane, some of your triangles

have sides that are of length –x or –y. Is the relationship still true for these

triangles? Why or why not?

6. What would be the equation of the graph that is the set on all points that are 6” away from

the origin?

7. Is the point (0, -6) on the graph? How about the point (3, 5.193)? How can you tell?

8. If the graph is translated 3 units to the right and 2 units up, what would be the equation of

the new graph? Explain how you found the equation.

4

NAME Circles and Other Conics 8.1

© 2013 MATHEMATICS VISION PROJECT | MVP In partnership with the Utah State Office of Education

Licensed under the Creative Commons Attribution-‐NonCommercial-‐ShareAlike 3.0 Unported license

Ready, Set, Go!

Ready Topic: Special products and factors Factor the following as the difference of 2 squares or as a perfect square trinomial. Do not factor if they are neither. 1. 25b2 − 49y2 2. 100b2 − 20b +1 3. 36b2 + 30b + 25 4. x2 + 6x + 9 − y2 5. x2 − 2xy + y2 − 25 6. a2 + 2ab + b2 + 4a + 4b + 4 7. x2 + 2xy +12x + y2 +12y + 36 8. x2 + 2cx + 2dx + c2 + 2cd + d 2 9. 144x2 − 312xy +169y2

Set Topic: Writing the equations of circles. Write the equation of each circle centered at the origin. 10.

11.

12.

© 2013 http://flic.kr/p/5m

EpJm

5




13.

14.

15.

Go Topic: Verifying Pythagorean triples. Identify which sets of numbers could be the sides of a right triangle. Show your work. 16. 9,12,15{ } 17. 9,10, 19{ } 18. 1, 3, 2{ } 19. 2, 4, 6{ } 20. 3, 4, 5{ } 21. 10, 24, 26{ }

22. 2, 7,3{ } 23. 2 2, 5 3, 9{ } 24. 4ab3 10, 6ab3, 14ab3{ }

6



8.2 Getting Centered A Solidify Understanding Task

Malik’s family has decided to put in a new sprinkling system in

their yard. Malik has volunteered to lay the system out.

Sprinklers are available at the hardware store in the following

sizes:

Full circle, maximum 15’ radius

Half circle, maximum 15’ radius

Quarter circle, maximum 15’ radius

All of the sprinklers can be adjusted so that they spray a smaller radius. Malik needs to be

sure that the entire yard gets watered, which he knows will require that some of the circular

water patterns will overlap. He gets out a piece of graph paper and begins with a scale

diagram of the yard. In this diagram, the length of the side of each square represents 5 feet.

ww

w.f

lickr

.co

m/p

ho

tos/

pay

no

min

d/1

48

47

65

71

lawn

7



1. As he begins to think about locating sprinklers on the lawn, his parents tell him to try

to cover the whole lawn with the fewest number of sprinklers possible so that they can

save some money. The equation of the first circle that Malik draws to represent the

area watered by the sprinkler is:

Draw this circle on the diagram using a compass.

2. Lay out a possible configuration for the sprinkling system that includes the first

sprinkler pattern that you drew in #1.

3. Find the equation of each of the full circles that you have drawn.

Malik wrote the equation of one of the circles and just because he likes messing with the

algebra, he did this:

Original equation:

Malik thought, “That’s pretty cool. It’s like a different form of the equation. I guess that there

could be different forms of the equation of a circle like there are different forms of the

equation of a parabola or the equation of a line.” He showed his equation to his sister, Sapana,

and she thought he was nuts. Sapana, said, “That’s a crazy equation. I can’t even tell where

the center is or the length of the radius anymore.” Malik said, “Now it’s like a puzzle for you.

I’ll give you an equation in the new form. I’ll bet you can’t figure out where the center is.”

8



Sapana said, “Of course, I can. I’ll just do the same thing you did, but work backwards.”

4. Malik gave Sapana this equation of a circle:

Help Sapana find the center and the length of the radius of the circle.

5. Sapana said, “Ok. I made one for you. What’s the center and length of the radius for

this circle?”

6. Sapana said, “I still don’t know why this form of the equation might be useful. When

we had different forms for other equations like lines and parabolas, each of the various

forms highlighted different features of the relationship.” Why might this form of the

equation of a circle be useful?

9




Ready, Set, Go! Ready Topic: Making perfect square trinomials. Fill in the number that completes the square. Then write the trinomial in factored form. 1. x2 + 6x + _________ 2. x2 −14x + _________

3. x2 − 50x + _________ 4. x2 − 28x + _________ On the next set of problems, leave the number that completes the square as a fraction. Then write the trinomial in factored form. 5. x2 −11x + _________ 6. x2 + 7x + _________ 7. x2 +15x + _________

8. x2 + 23x + _________ 9. x2 − 1

5x + _________ 10. x2 − 3

4x + _________

Set Topic: Writing equations of circles with center (h, k) and radius r. Write the equation of the circle. 11. 12. 13.

©2013www.flickr.com/photos/paynom

ind/148476571

10




Write the equation of the circle with the given center and radius. Then write it in expanded form. 14. center (5, 2) radius 13 15. center (-‐6, -‐10) radius 9 16. center (0, 8) radius 15 17. center (19, -‐13) radius 1 18. center (-‐1, 2) radius 10 19. center (-‐3, -‐4) radius 8

Go Topic: Verifying if a point is a solution. Identify which point is a solution to the given equation. Show your work.

20. y = 45x − 2 A. (-‐15, -‐14) B. ( 10, 10)

21. y = 3 x A. ( -‐4, -‐12) B. − 5,3 5( ) 22. y = x2 + 8 A. 7,41( ) B. 7,−1( ) 23. y = −4x2 +120 A. 5 3,−180( ) B. 5 3,40( ) 24. x2 + y2 = 9 A. 8,−1( ) B. −2, 5( ) 25. 4x2 − y2 = 16 A. −3, 10( ) B. −2 2,4( )

11



8.3 Circle Challenges A Practice Understanding Task

Once Malik and Sapana started challenging each other with

circle equations, they got a little more creative with their

ideas. See if you can work out the challenges that they gave

each other to solve. Be sure to justify all of your answers.

1. Malik’s challenge:

What is the equation of the circle with center (-13,-16) and containing the point (-10,-16) on

the circle?

2. Sapana’s challenge:

The points (0, 5) and (0,-5) are the endpoints of the diameter of a circle. The point (3, y) is

on the circle. What is a value for y?


Find the equation of a circle with center in the first quadrant and is tangent to the lines

x = 8, y = 3, and x = 14.


The points (4,-1) and (-6,7) are the endpoints of the diameter of a circle. What is the

equation of the circle?

ww

w.f

lickr

.co

m/p

ho

tos/

osc

hen

e

12




Is the point (5,1) inside, outside, or on the circle ? How do you

know?


The circle defined by is translated 5 units to the left and 2

units down. Write the equation of the resulting circle.


There are two circles, the first with center (3,3) and radius r1, and the second with center

(3, 1) and radius r2. a. Find values r1 and r2 of so that the first circle is completely enclosed by the second

circle.

b. Find one value of r1 and one value of r2 so that the two circles intersect at two points.

c. Find one value of r1 and one value of r2 so that the two circles intersect at exactly one

point.

13




Ready, Set, Go!

Ready Topic: Finding the distance between 2 points.

Use the distance formula d = x2 − x1( )2 + y2 − y1( )2 to find the distance between the given points. Leave your answer in simplest radical form. 1. A 18,−12( ) B 10,4( )

2. G −11,−9( ) H −3,7( ) 3. J 14,−20( ) K 5,5( ) 4. M 1,3( ) P −2,7( )

5. Q 8,2( ) R 3,7( ) 6. S −11,2 2( ) T −5,−4 2( ) 7. W −12,−2 2( ) Z −7,−3 2( )

Set Topic: Writing equations of circles Use the information provided to write the equation of the circle in standard form. 8. Center −16,−5( ) and the circumference is 22π

9. Center 13,−27( ) and the area is 196π

10. Diameter measures 15 units and the center is at the intersection of y = x + 7 and y = 2x – 5

11. Lies in quadrant 2 Tangent to x = −12 and x = −4

©2013www.flickr.com/photos/oschene

14




12. Center −14,9( ) Point on circle 1,11( )

13. Center lies on the y axis Tangent to y = -‐2 and y = -‐17

14. Three points on the circle are −8,5( ), 3,−6( ), 14,5( )

15. I know three points on the circle are (-‐7,6), (9,6), and (-‐4,13). I think that the equation of the circle is x −1( )2 + y − 6( )2 = 64 . Is this the correct equation for the circle? Convince me!

Go Topic: Finding the value of “B” in a quadratic of the form Ax2 + Bx +C in order to create a perfect square trinomial. Find the value of “B,” that will make a perfect square trinomial. Then write the trinomial in factored form. 16. x2 + _____ x + 36 17. x2 + _____ x +100 18. x2 + _____ x + 225 19. 9x2 + _____ x + 225 20. 16x2 + _____ x +169 21. x2 + _____ x + 5

22. x2 + _____ x + 254

23. x2 + _____ x + 94 24. x2 + _____ x + 49

4

15



8.4 Directing Our Focus A Develop Understanding Task

On a board in your classroom, your teacher has set up

a point and a line like this:

Focus (point A)

directrix (line l)

We’re going to call the line a directrix and the point a focus. They’ve been labeled on the drawing.

Similar to the circles task, the class is going to construct a geometric figure using the focus (point A)

and directrix (line l).

1. Cut two pieces of string with the same length.

2. Mark the midpoint of each piece of string with a marker.

ww

w.f

lickr

.co

m/p

ho

tos/

pix

elth

ing

16

http://www.flickr.com/photos/pixelthing



3. Position the string on the board so that the midpoint is equidistant from the focus (point A)

and the directrix (line l) While

holding the string in this position, put a pin through the midpoint. Depending on the size of

your string, it will look something like this:

4. Using your second string, use the same procedure to post a pin on the other side of the

focus.

5. As your classmates post their strings, what geometric figure do you predict will be made by

the tacks (the collection of all points like (x, y) show in the figure above)? Why?

6. Where is the vertex of the figure located? How do you know?

7. Where is the line of symmetry located? How do you know?

(x, y)

17



8. Consider the following construction with focus point A and the x axis as the directrix. Use a

ruler to complete the construction of the parabola in the same way that the class

constructed the parabola with string.

9. You have just constructed a parabola based upon the definition: A parabola is the set of all

points (x, y) equidistant from a line l (the directrix) and a point not on the line (the focus).

Use this definition to write the equation of the parabola above, using the point (x, y) to

represent any point on the parabola.

10. How would the parabola change if the focus was moved up, away from the directrix?

11. How would the parabola change if the focus was moved down, toward the directrix?

12. How would the parabola change if the focus was moved down, below the directrix?

(x, y)

18




Ready, Set, Go! Ready Topic: Graphing quadratics Graph each set of functions on the same coordinate axes. Describe in what way the graphs are the same and in what way they are different.

1. y = x2, y = 2x2, y = 4x2 2. y = 14x2, y = −x2, y = −4x2

3. y = 14x2, y = x2 − 2, y = 1

4x2 − 2, y = 4x2 − 2 4. y = x2, y = −x2, y = x2 + 2, y = −x2 + 2

Set Topic: Sketching a parabola using the conic definition.

©2013www.flickr.com/photos/pixelthing

19




Use the conic definition of a parabola to sketch a parabola defined by the given focus F and the equation of the directrix. Begin by graphing the focus, the directrix, and point P1 . Use the distance formula to find FP1 and find the vertical distance between P1 and the directrix by identifying point H on the directrix and counting the distance. Locate the point P2 , (the point on the parabola that is a reflection of P1 across the axis of symmetry.) Locate the vertex V. Since the vertex is a point on the parabola, it must lie equidistant between the focus and the directrix. Sketch the parabola. Hint: the parabola always “hugs” the focus.

Example: F (4,3), P1 (8, 6), y = 1 FP1 = 4 − 8( )2 + 3− 6( )2 = 16 + 9 = 25 = 5 P1H = 5 P2 is located at (0, 6) V is located at (4, 2) 5. F (1,-‐1), P1 (3, -‐1) y = -‐3 6. F (-‐5,2), P1 (-‐9, -‐1) y = 7 7. F (2,2), P1 (-‐2, 5) y =-‐3 8. F (1, -‐1), P1 (-‐7, -‐7) y = 9

20




9. Find a square piece of paper (a post-‐it note will work). Fold the square in half vertically and put a dot anywhere on the fold. Let the edge of the paper be the directrix and the dot be the focus. Fold the edge of the paper (the directrix) up to the dot repeatedly from different points along the edge. The fold lines between the focus and the edge should make a parabola. Experiment with a new paper and move the focus. Use your experiments to answer the following questions. 10. How would the parabola change if the focus were moved up, away from the directrix? 11. How would the parabola change if the focus were moved down, toward the directrix? 12. How would the parabola change if the focus were moved down, below the directrix?

Go Topic: Finding the center and radius of a circle. Write each equation in standard form. Find the center (h, k) and radius r of the circle. 13. x2 + y2 + 4y −12 = 0 14. x2 + y2 − 6x − 3= 0 15. x2 + y2 + 8x + 4y − 5 = 0 16. x2 + y2 − 6x −10y − 2 = 0 17. x2 + y2 − 6y − 7 = 0 18. x2 + y2 − 4x + 8y + 6 = 0 19. x2 + y2 − 4x + 6y − 72 = 0 20. x2 + y2 +12x + 6y − 59 = 0 21. x2 + y2 − 2x +10y + 21= 0 22. 4x2 + 4y2 + 4x − 4y −1= 0

21



8.5 Functioning With Parabolas A Solidify Understanding Task

Sketch the graph (accurately), find the vertex and use the

geometric definition of a parabola to find the equation of

these parabolas.

1. Directrix y = -4, Focus A(2, -2)

Vertex ________________

Equation:

2. Directrix y = 2, Focus A(-1, 0)

Vertex ________________

Equation:

©2

01

3 w

ww

.flic

kr.c

om

/ph

oto

s/m

ore

than

mat

hs/

22

62

193

46

1

22



3. Directrix y = 3, Focus A(1, 7)

Vertex ________________

Equation:

4. Directrix y = 3, Focus A(2, -1)

Vertex ________________

Equation:

5. Given the focus and directrix, how can you find the vertex of the parabola?

23



6. Given the focus and directrix, how can you tell if the parabola opens up or down?

7. How do you see the distance between the focus and the vertex (or the vertex and the

directrix) showing up in the equations that you have written?

8. Describe a pattern for writing the equation of a parabola given the focus and directrix.

9. Annika wonders why we are suddenly thinking about parabolas in a completely

different way than when we did quadratic functions. She wonders how these different

ways of thinking match up. For instance, when we talked about quadratic functions

earlier we started with . “Hmmmm. …. I wonder where the focus and directrix

would be on this function,” she thought. Help Annika find the focus and directrix for

.

24



10. Annika thinks, “Ok, I can see that you can find the focus and directrix for a quadratic

function, but what about these new parabolas. Are they quadratic functions? When we

work with families of functions, they are defined by their rates of change. For instance,

we can tell a linear function because it has a constant rate of change.” How would you

answer Annika? Are these new parabolas quadratic functions? Justify your answer

using several representations and the parabolas in problems 1-4 as examples.

25




Ready, Set, Go!

Ready Topic: Standard form of a quadratic. Verify that the given point lies on the graph of the parabola described by the equation. (Show your work.) 1. 6,0( ) y = 2x2 − 9x −18

2. −2,49( ) y = 25x2 + 30x + 9 3. 5,53( ) y = 3x2 − 4x − 2 4. 8,2( ) y = 14x2 − x − 6

Set Topic: the equation of a parabola based on the geometric definition

5. Verify that y −1( ) = 14x2 is the equation

of the parabola in figure 1 by plugging in the 3 points V (0,1), C (4,5) and E (2,2). Show your work for each point! 6. If you didn’t know that (0,1) was the vertex of the parabola, could you have found it by just looking at the equation? Explain.

Figure 1

©2013www.flickr.com/photos/morethanm

aths/226219461

26




7. Use the diagram in figure 2 to derive the general equation of a parabola based on the geometric definition of a parabola. Remember that the definition states that PF = PQ.

8. Recall the equation in #5, y −1( ) = 14x2 , what is the value of a?

9. In general, what is the value of a in any parabola? 10. In figure 3, the point M is the same height as the focus and FM ≅ MR . How do the coordinates of this point compare with the coordinates of the focus? Fill in the missing coordinates for M and R in the diagram.

figure 2

figure 3

27




Sketch the graph by finding the vertex and the point M and M’ (the reflection of M) as defined in the diagram above. Use the geometric definition of a parabola to find the equation of these parabolas. 11. Directrix y = 9, Focus A(-‐3, 7) 12. Directrix y = -‐6, Focus A(2, -‐2) Vertex _____________Equation _______________ Vertex _____________Equation _______________

13. Directrix y = 5, Focus A(-‐4, -‐1) 14. Directrix y = -‐1, Focus A(4, -‐3) Vertex _____________Equation _______________ Vertex _____________Equation _______________

28




Go Find the maximum or minimum value of the quadratic. Indicate which it is. 15. y = x2 + 6x − 5 16. y = 3x2 −12x + 8

17. y = − 12x2 +10x +13 18. y = −5x2 + 20x −11

19. y = 72x2 − 21x − 3 20. y = − 3

2x2 + 9x + 25

29



8.6 Turn It Around A Solidify Understanding Task

Annika is thinking more about the geometric view of

parabolas that she has been working on in math class.

She thinks, “Now I see how all the parabolas that come

from graphing quadratic functions could also come from a given focus and directrix. I notice that all

the parabolas have opened up or down when the directrix is horizontal. I wonder what would

happen if I rotated the focus and directrix 90 degrees so that the directrix is vertical. How would

that look? What would the equation be? Hmmm….” So Annika starts trying to construct a parabola

with a vertical directrix. Here’s the beginning of her drawing. Use a ruler to complete Annika’s

drawing.

1. Use the definition of a parabola to write the equation of Annika’s parabola.

© 2

01

3 w

ww

.flic

kr.c

om

/ph

oto

s/ro

bw

alla

ce 30



2. What similarities do you see to the equations of parabolas that open up or down? What

differences do you see?

3. Try another one: Write the equation of the parabola with directrix x = 4 and focus (0, 3).

4. One more for good measure: Write the equation of the parabola with directrix x = -3 and

focus (-2, -5).

5. How can you predict if a parabola will open left, right, up, or down?

6. How can you tell how wide or narrow a parabola is?

7. Annika has two big questions left. Write and explain your answers to these questions.

a. Are parabolas functions?

b. Are all parabolas similar?

31




Ready, Set, Go!

Ready Topic: Are you ready for a test on module 8?

Review of circles Use the given information to write the equation of the circle in standard form.

1. center (-‐5, -‐8) radius 11

2. endpoints of the diameter (6, 0) and (2, -‐4)

3. center (-‐5, 4) point on the circle (-‐9, 1)

4. equation of the circle in the diagram.

Set Topic: Writing equations of horizontal parabolas Use the focus F, point P, a point on the parabola, and the equation of the directrix to sketch the parabola (label your points) and write the equation. Put your equation in the form

x − h( ) = 14a

y − k( )2 , where "a" is the dis tance from the focus to the vertex.

5. F (1,0), P (1,4) x = -‐3 6. F (3,1), P (2,-‐5) x = 9

© 2013 www.flickr.com/photos/robw

allace

32




7. F (7,-‐5), P (4,3) x = 9 8. F (-‐1,2), P (8,-‐10) x = -‐7

Go Topic: Identifying key features of a quadratic written in vertex form State (a) the coordinates of the vertex, (b) the equation of the axis of symmetry, (c) the domain, and (d) the range for each of the following functions. 9. f x( ) = x − 3( )2 + 5 10. f x( ) = x +1( )2 − 2 11. f x( ) = − x − 3( )2 − 7

12. f x( ) = −3 x − 34

⎛⎝⎜

⎞⎠⎟2

+ 45 13. f x( ) = 1

2x − 4( )2 +1 14. f x( ) = 1

4x + 2( )2 − 4

15. Compare the vertex form of a quadratic to the geometric equation of a parabola. Describe how they are similar and how they are different.

33

Documents

Secondary Two Mathematics: An Integrated …...Circles and Other Conics 8.1 Classroom Task: 8.2 Getting Centered – A Solidify Understanding Task Complete the square to find the center